Exercise 13, § 1

Let E be an ordered set. A mapping $f$ of E into itself is said to be a closure if it satisfies the following conditions : (1) $f$ is increasing; (2) for each $x \in E$, $f(x) \geqslant x$; (3) for each $x \in E$, $f(f(x)) = f(x)$. Let F be the set of elements of E which are invariant under $f$.

(a) Show that for each $x \in E$ the set $F_x$ of elements $y \in F$ such that $x \leqslant y$ is not empty and has a least element, namely $f(x)$. Conversely, if G is a subset of E such that, for each $x \in E$, the set of all $y \in G$ such that $x \leqslant y$ has a least element $g(x)$, then $g$ is a closure and G is the set of elements of E which are invariant under $g$.

(b) Suppose that E is a complete lattice. Show that the greatest lower bound in E of any non-empty subset of F belongs to F.

(c) Show that if E is a lattice, then $f(\sup (x, y)) = f(\sup (f(x), y))$ for each pair of elements $x, y$ of E.